NotImplementedError:
This equation is not yet recognized or else has not been simplified
sufficiently to put it in a form recognized by diop_classify().
sage: version()
'SageMath version 8.1, Release Date: 2017-12-07'
sage:

which seems perfectly valid. That said, if you do not care about the integer nature of the solution, there is no benefit in using of a computer algebra system to solve such an equation since the variable c is isolated, it is clear that for each a and each b, there are two solutions sqrt(a^2 + b^2 + 2*a^2*b^2) and sqrt(a^2 + b^2 + 2*a^2*b^2).

The question is when a^2 + b^2 + 2*a^2*b^2) is a square. You can try to guess some pattern for the integer solutions as follows (try with size=10,100,1000,10000):